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Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

An Approach to Mathematical Finance

David Ruiz

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Table of contents

1 Financial Market 2 The Market Model 3 Portfolio and Derivatives 4 Pricing Theory 5 Risk Minimization

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Financial Market

Definition (Financial Market) A Financial Market is a market in which people and companies can trade financial securities, commodities, stocks or other equities or assets. Example Financial securities include stocks or bonds. A market of commodities include metals, oil, agricultural goods, ...

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Stochastic process

Definition (Random variable) Given a probability space (Ω, F, P), with Ω a sample space, F its σ-algebra of events and P a probability. A random variable is a measurable function or map X : (Ω, F) → (R, B(R)). Definition (Stochastic process) A stochastic process is a function X : [0, T] × Ω → R such that, for all fixed t ∈ [0, T], X(t, ·) is a random variable and for all fixed ω ∈ Ω, X(·, ω) is a real-valued ordinary function. So, a stochastic process is simply a family (countable or uncountable) of random variables.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Example

Denote by St(ω) =”the price of a share of telefonica at a given time t ∈ [0, T], fixed ω ∈ Ω”.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Example

Denote by St(ω) =”the price of a share of telefonica at a given time t ∈ [0, T], fixed ω ∈ Ω”. So, for intance, given an event {ω0} ∈ F, we have St(ω0) = et (the price goes up).

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Example

Denote by St(ω) =”the price of a share of telefonica at a given time t ∈ [0, T], fixed ω ∈ Ω”. So, for intance, given an event {ω0} ∈ F, we have St(ω0) = et (the price goes up). Given another completely different event {ω1} ∈ F, we have St(ω1) = −et (the price goes down).

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Example

Denote by St(ω) =”the price of a share of telefonica at a given time t ∈ [0, T], fixed ω ∈ Ω”. So, for intance, given an event {ω0} ∈ F, we have St(ω0) = et (the price goes up). Given another completely different event {ω1} ∈ F, we have St(ω1) = −et (the price goes down). Also, fixed t, let’s say t =”tomorrow”, we can have, St ∼ N(12, 2.5).

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Example

Denote by St(ω) =”the price of a share of telefonica at a given time t ∈ [0, T], fixed ω ∈ Ω”. So, for intance, given an event {ω0} ∈ F, we have St(ω0) = et (the price goes up). Given another completely different event {ω1} ∈ F, we have St(ω1) = −et (the price goes down). Also, fixed t, let’s say t =”tomorrow”, we can have, St ∼ N(12, 2.5). In reality, St(ω) is not deterministic, but random.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Martingale Process

We say that a stochastic process Mt(ω) is a P-martingale if (integrable + adapted) E(Mt|Fs) = Ms for all t s.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Martingale Process

We say that a stochastic process Mt(ω) is a P-martingale if (integrable + adapted) E(Mt|Fs) = Ms for all t s. Example: Mt(ω) represents the income or benefit from playing a game.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Martingale Process

We say that a stochastic process Mt(ω) is a P-martingale if (integrable + adapted) E(Mt|Fs) = Ms for all t s. Example: Mt(ω) represents the income or benefit from playing a game. E(Mt|Fs) Ms Submartingale → Game favouring player.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Martingale Process

We say that a stochastic process Mt(ω) is a P-martingale if (integrable + adapted) E(Mt|Fs) = Ms for all t s. Example: Mt(ω) represents the income or benefit from playing a game. E(Mt|Fs) Ms Submartingale → Game favouring player. E(Mt|Fs) = Ms Martingale → Fair game.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Martingale Process

We say that a stochastic process Mt(ω) is a P-martingale if (integrable + adapted) E(Mt|Fs) = Ms for all t s. Example: Mt(ω) represents the income or benefit from playing a game. E(Mt|Fs) Ms Submartingale → Game favouring player. E(Mt|Fs) = Ms Martingale → Fair game. E(Mt|Fs) Ms Supermartingale → Game favouring counterpart.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

The Market Model

We consider n + 1, financial securities, let’s say stocks, S0

t (ω), S1 t (ω), . . . , Sn t (ω), where the first one is risk-less and the

• ther n are risky.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

The Market Model

The risk-less financial security, measured in domestic currency, is driven by the following stochastic differential equation

• dS0

t (ω) = S0 t (ω)r(t, ω)dt,

S0

0(ω) = 1.

Here, r is called the interest rate and it does not need to be

• deterministic. This could, for instance, be a bank account or bond,

although bonds may default.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

The Market Model

The other n stocks are driven by the following SDE

• dSi

t(ω) = Si t(ω)(bi(t, ω)dt + n k=1 σi,k(t, ω)dW i t (ω)),

Si

0(ω) > 0,

where Si

t is the i-th security asset.

Wt is a (vector) stochastic process with infinite variation. dWt is a ”kind of differential”, a noise. b is called the drift (a vector) and σ is the volatility (a matrix).

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

The Market Model

Example of modelling stock prices using brownian motion.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Portfolio

Definition (Portfolio) A portfolio or strategy is a vector stochastic process θ = (θ0

t (ω), θ1 t (ω), . . . , θn t (ω)), where each θi t(ω) denotes the

number of units invested in the stock i at time t ∈ [0, T], and i = 0, . . . , n. Definition The value of a portfolio denoted by V θ

t (ω) is given by the discrete

scalar product, V θ

t (ω) = n

• i=0

θi

t(ω)Si t(ω) = θ · S

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Portfolio

Definition (Admissibility) A portfolio θ is said to be admissible if its value is almost surely non-negative, i.e.: V θ

t (ω) 0, P-a.s.

Definition (Self-financing portfolio) We say that a portfolio θ is self-financing if dV θ

t = θ0 t dS0 t + n i=1 θi tdSi t.

Hence also, Vt = V0 + t θ0

s dS0 s + n

• i=1

t θi

sdSi s.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Arbitrage opportunity

Definition (Arbitrage opportunity) A portfolio or strategy θ is said to be an arbitrage opportunity if V θ

0 = 0 and there exists a time t ∈ [0, T] such that V θ t > 0, with

strictly positive probability. That is, P(ω : V θ

t (ω) > 0) > 0.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Derivative

A derivative or option is a contract where the seller of this contract gets an amount of money from the buyer of this contract. Definition (Contingent Claim) A contingent claim is an FT-measurable, and positive random variable. Example: A European call option.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Fair price

Imagine we have an option with a final payoff h(ω) at maturity time T. How much should the buyer of this option pay now? Definition (Replicating portfolio) Consider a contingent claim or payoff h. A replicating or hedging portfolio for h is a self-financing portfolio θ such that its final value is h, i.e.: V θ

T(ω) = h(ω), P-a.s.

Example: A European call option. The buyer of this option gets h = (S(T) − K)+.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Fair price

The price of a contract now (t = 0) with payoff h at time T is the initial value of a portfolio such that it replicates h. Fair price now = V θ with V θ

T = h.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Fair price

The price of a contract now (t = 0) with payoff h at time T is the initial value of a portfolio such that it replicates h. Fair price now = V θ with V θ

T = h.

Definition (Complete Market) A financial market is said to be complete, if any contingent claim is replicable.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Arbritrage opportunity example

A market bond/stock Asset/Time Today t=0 Tomorrow t=1 Bond Bt 0.95 $ 1 $ Stock St 2 $ ω = ω1 1 $ 1/2 $ ω = ω2 What is the fair price of European call C1 = (S1 − K)+ with strike K = 1?

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Arbritrage opportunity example

Example: A car in USA is cheaper than in Canada. Americans: Buy car in USA, drive to Canada, sell more expensive and sell CAD they get for the sale. Canadians: Buy USD to buy a car in USA and return home. The exploit of this arbitrage makes supply of CAD and demand for USD increase. This implies USD increase in price and CAD decrease, this would make the price of American cars more expensive until the arbitrage situation disappears.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

A Pricing Formula

pt = S0

t EP∗

h S0

T

|Ft

• ,

where P∗ is a different probability, equivalent to P (the risk neutral probability).

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

A Pricing Formula

pt = S0

t EP∗

h S0

T

|Ft

• ,

where P∗ is a different probability, equivalent to P (the risk neutral probability). The factor we use to discount is called num´ eraire or benchmark. There are other formulas changing measure and num´ eraire.

David Ruiz An Approach to Mathematical Finance

Pricing Theory

(A) Robert Merton (born 31 July 1944, aged 67): American economist. (C) Myron Scholes (born July 1, 1941, aged 70): Canadian-born American financial economist. (B) Fischer Black (January 11, 1938 – August 30, 1995): American economist.

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Risk Minimization

Let h be a claim which is not attainable. This means that an amissible portfolio θ whose final value is: V θ

T(ω) = h(ω)

can not be self-financing. So, it has a cost

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Risk Minimization

Not all markets are complete! i.e.: Not all claims are replicable or attainable.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Risk Minimization

Not all markets are complete! i.e.: Not all claims are replicable or attainable. The cost of a non-self-financing strategy at time t ∈ [0, T] is C θ

t (ω) = V θ t (ω) − V θ 0 (ω) −

t θs(ω) · dSs(ω) which represents the part of the value that has not been gained from trading.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Risk Minimization

Definition (Risk) Given a non-self-financing strategy θ, and C its cost process, we define the risk of this strategy at time t ∈ [0, T] as Rt = E

• C T − C t

2 |Ft

• ,

where C = C

S0 . This is, the mean square of its remaining cost.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Risk Minimization

Substituting we have that the risk of a non self-financing strategy is: Rt = E

• h − V

θ t −

T

t

θsdSs 2 |Ft

• which can be seen as a way to measure how well the current value
• f the portfolio plus future trading can approximate the

non-attainable claim.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

References

Karatzas, Ioannis, Shreve, Steven E..; Methods of Mathematical Finance. Springer 1998. Ragnar Norberg. Basic Life Insurance Mathematics. Notes 2002. Bernt Øksendal. Stochastic Differential Equations. Springer.

David Ruiz An Approach to Mathematical Finance

Financial Market The Market Model Portfolio and Derivatives Pricing Theory Risk Minimization

Thank you!

David Ruiz An Approach to Mathematical Finance